Recall that our method (see Eq. REF ) enforces the distribution of task-independent factors to approximate a standard Gaussian: \((S_{\hat{z}}|S_z) = S_{\hat{z}} \rightarrow \mathcal {N}(0; I)\)  and \(T_{\hat{z}} \rightarrow \mathcal {N}(0; I)\) , hence enforcing the last integral term of Eq. REF : \(d_{\mathcal {H}\Delta \mathcal {H}}(S_{\hat{z}}|S_z, T_{\hat{z}}) \rightarrow d_{\mathcal {H}\Delta \mathcal {H}}(\mathcal {N}, \mathcal {N}) = 0\) . Consequently, our method attempts to optimize
the upper bound for \(d_{\mathcal {H}\Delta \mathcal {H}}(S, T)\) , thus offering a tighter upper bound for \(\epsilon ^T(h)\) . The proof shows that our method is mathematically consistent with the theory of Ben-David et al [1]}.
